A Treatise on the Theory of Screws

242] FBEEDOM OF THE SIXTH ORDER. 261 Expressing the condition that pe = 0, we have tftpW + 2hklplV1 ~ = 0 ; dy1 Jh \dV1J but we have already seen (§ 131) that the two last terms of this equation are zero, whence the required theorem is demonstrated. The formula we have just proved may be written in the form S/>!. p^i. p^ = 0. This shows that if the body were free, then an impulsive force suitably placed would make the body commence to rotate about y. Whence we have the following theorem*:— A rigid body previously in unconstrained equilibrium in free space is supposed to be set in motion by a single impulsive force; if the initial axis of twist velocity be a principal axis of the body, the initial motion is a pure rotation, and conversely. It may also be asked at what point of the body one of the three principal axes coincides with y ? This point is the intersection of 6 and y. To determine the co-ordinates of 0 it is only necessary to find the relation between h and k, and this is obtained by expressing the condition that 0 is reciprocal to y, whence we deduce 2h + kvf — 0. Thus 6 is known, and the required point is determined. If the body be fixed at this point, and then receive the impulsive couple perpendicular to y, the instantaneous reaction of the point will be directed along 0. 242. Harmonic Screws. We shall conclude by stating for the sixth order the results which are included as particular cases of the general theorems in Chapter IX. If a perfectly free rigid body be in equilibrium under the influence of a conservative system of forces, then six screws can generally be found such that each pair are conjugate screws of inertia, as well as conjugate screws of the potential, and these six screws are called harmonic screws. If the body be displaced from its position of equilibrium by a twist of small amplitude about a harmonic screw, and if the body further receive a small initial twisting motion about the same screw, then the body will continue for ever to perform small twist oscillations about that screw. And, more generally, whatever be the initial circumstances, the movement of the body is com- pounded of twist oscillations about the six harmonic screws. Townsend, Educational Times Reprint, Vol. xxi. p. 107.